Find Second Order Partial Derivative Calculator

Calculate second-order partial derivatives (fxx, fyy, fxy, fyx) for the function f(x,y) = Ax^ay^b + Cx^c + Dy^d + E at a point (x, y).

Chart showing fxx and fyy as x varies around 1 (y=1)

What is a Second Order Partial Derivative?

A second order partial derivative is a derivative of a function of multiple variables with respect to one variable, taken twice, or with respect to two different variables sequentially. For a function f(x, y), we have four second-order partial derivatives: f_xx (∂²f/∂x²), f_yy (∂²f/∂y²), f_xy (∂²f/∂x∂y), and f_yx (∂²f/∂y∂x).

f_xx means differentiating f with respect to x, and then differentiating the result with respect to x again. f_xy means differentiating f with respect to x, and then differentiating the result with respect to y. Under certain continuity conditions (Clairaut’s theorem), f_xy = f_yx.

The second order partial derivative calculator helps compute these values for a given function and point, which are crucial in multivariable calculus for tasks like finding local extrema (using the second derivative test) and understanding the curvature of a surface.

Who should use it?

Students studying multivariable calculus, engineers, physicists, economists, and anyone dealing with functions of multiple variables will find a second order partial derivative calculator useful. It’s particularly helpful for checking manual calculations and for quickly evaluating derivatives at specific points.

Common misconceptions

A common misconception is that f_xy is always different from f_yx. However, if the second partial derivatives are continuous in a region, Clairaut’s theorem states that f_xy = f_yx in that region. Our second order partial derivative calculator calculates both, and for the polynomial-like functions used here, they are indeed equal.

Second Order Partial Derivative Formula and Mathematical Explanation

For a general function f(x, y), the second order partial derivatives are defined as:

• f_xx = ∂²f/∂x² = ∂/∂x (∂f/∂x): Differentiate f with respect to x, then differentiate the result with respect to x.
• f_yy = ∂²f/∂y² = ∂/∂y (∂f/∂y): Differentiate f with respect to y, then differentiate the result with respect to y.
• f_xy = ∂²f/∂y∂x = ∂/∂y (∂f/∂x): Differentiate f with respect to x, then differentiate the result with respect to y.
• f_yx = ∂²f/∂x∂y = ∂/∂x (∂f/∂y): Differentiate f with respect to y, then differentiate the result with respect to x.

For the function used in our second order partial derivative calculator, f(x,y) = Ax^ay^b + Cx^c + Dy^d + E:

First derivatives:

• ∂f/∂x = Aax^a-1y^b + Ccx^c-1
• ∂f/∂y = Abx^ay^b-1 + Ddy^d-1

Second derivatives:

• f_xx = ∂²f/∂x² = A*a*(a-1)x^a-2y^b + C*c*(c-1)x^c-2
• f_yy = ∂²f/∂y² = A*b*(b-1)x^ay^b-2 + D*d*(d-1)y^d-2
• f_xy = ∂²f/∂y∂x = A*a*bx^a-1y^b-1
• f_yx = ∂²f/∂x∂y = A*b*ax^a-1y^b-1

Variables Table

Table of variables used in the second order partial derivative calculation.

Variable	Meaning	Unit	Typical Range
A, C, D, E	Coefficients and constant in the function	Dimensionless (depends on context)	Any real number
a, b, c, d	Exponents in the function	Dimensionless	Any real number (often integers or simple fractions)
x, y	Independent variables	Depends on context	Any real number
fxx, fyy, fxy, fyx	Second order partial derivatives	Depends on context	Any real number

Practical Examples (Real-World Use Cases)

Example 1: Finding Curvature

Suppose the height of a hill is given by h(x,y) = -0.1x² – 0.2y² + 500, where x and y are distances east and north from a reference point. We want to find the curvature at (x=10, y=5).
Using our calculator’s structure, this is like f(x,y) with A=0, C=-0.1, c=2, D=-0.2, d=2, E=500.

h_x = -0.2x, h_y = -0.4y

h_xx = -0.2 (constant)

h_yy = -0.4 (constant)

h_xy = 0

At (10, 5), h_xx = -0.2, h_yy = -0.4, h_xy = 0. The negative values indicate the surface is concave down in both x and y directions, suggesting a peak.

Example 2: Second Derivative Test

Let f(x,y) = x²y – 2xy + y² – 3y. We find critical points where fx=0 and fy=0. Let’s say we found a critical point at (1, 2) and want to classify it using the second derivative test with our second order partial derivative calculator (or manual calculation for this different function).

f_x = 2xy – 2y => At (1,2), f_x = 4-4=0

f_y = x² – 2x + 2y – 3 => At (1,2), f_y = 1-2+4-3=0

f_xx = 2y => At (1,2), f_xx = 4

f_yy = 2 => At (1,2), f_yy = 2

f_xy = 2x – 2 => At (1,2), f_xy = 0

The discriminant D = f_xxf_yy – (f_xy)² = (4)(2) – 0² = 8. Since D > 0 and f_xx > 0, the point (1, 2) corresponds to a local minimum.

How to Use This Second Order Partial Derivative Calculator

1. Enter the Function Parameters: Input the values for coefficients A, C, D, E and exponents a, b, c, d for the function f(x,y) = Ax^ay^b + Cx^c + Dy^d + E.
2. Enter the Point of Evaluation: Input the x and y coordinates (xVal, yVal) at which you want to evaluate the derivatives.
3. Calculate: The calculator automatically updates the results as you type, or you can click “Calculate”.
4. Read Results: The “Results” section will display the values of f_xx, f_yy, f_xy, and f_yx at the specified point (xVal, yVal). The value of f_xx is highlighted.
5. View Chart: The chart shows how f_xx and f_yy change as x varies around your input xVal, keeping y constant at yVal.
6. Reset: Click “Reset” to return to default values.
7. Copy: Click “Copy Results” to copy the main results and input values to your clipboard.

The second order partial derivative calculator provides immediate feedback, making it easy to see how changes in the function or the point affect the derivatives.

Key Factors That Affect Second Order Partial Derivative Results

• Function Form: The specific algebraic form of f(x,y) (coefficients A, C, D, E and exponents a, b, c, d) directly dictates the form of its derivatives.
• Exponents (a, b, c, d): These determine the power to which x and y are raised and significantly influence how rapidly the function and its derivatives change. Higher exponents can lead to larger magnitude derivatives.
• Coefficients (A, C, D, E): These scale the terms and thus scale the derivatives proportionally.
• Point of Evaluation (xVal, yVal): The values of the derivatives depend on the point (x, y) at which they are evaluated, unless the derivatives are constant.
• Interaction between x and y (Term Ax^ay^b): The mixed term involving both x and y influences the mixed partial derivatives f_xy and f_yx.
• Continuity of Derivatives: For f_xy to equal f_yx, the second partial derivatives need to be continuous. For the polynomial-like functions used here, this is generally the case.

Understanding these factors is crucial when using a second order partial derivative calculator for analysis, like in optimization problems or when studying the local behavior of a function.

Frequently Asked Questions (FAQ)

What are second order partial derivatives used for?

They are used in the second derivative test to classify critical points (local max, min, or saddle points), in physics to describe wave equations and heat equations, and in economics for optimization problems.

Why is f_xy often equal to f_yx?

This is due to Clairaut’s Theorem (or Schwarz’s theorem or Young’s theorem), which states that if the second partial derivatives are continuous in a neighborhood of a point, then the order of differentiation does not matter (f_xy = f_yx).

Can this calculator handle any function f(x,y)?

No, this specific second order partial derivative calculator is designed for functions of the form f(x,y) = Ax^ay^b + Cx^c + Dy^d + E. More complex functions require symbolic differentiation tools.

What is the Hessian matrix?

The Hessian matrix is a square matrix of second-order partial derivatives of a scalar-valued function. For f(x,y), it is [[f_xx, f_xy], [f_yx, f_yy]]. Our second order partial derivative calculator finds the elements of this matrix.

How does the second derivative test work?

At a critical point (where f_x=0 and f_y=0), you evaluate D = f_xxf_yy – (f_xy)². If D>0 and f_xx>0, it’s a local min. If D>0 and f_xx<0, it's a local max. If D<0, it's a saddle point. If D=0, the test is inconclusive.

What if my exponents are not integers?

The formulas still apply, but you need to be careful about the domain of the function and its derivatives (e.g., x^1/2 is defined for x>=0).

Can I use negative exponents?

Yes, negative exponents like x^-1 = 1/x are valid, but ensure the function and its derivatives are defined at the point (x,y) you are evaluating (e.g., x cannot be 0 if x^-1 is present).

What does a zero second partial derivative mean?

For example, if f_xx=0, it suggests the rate of change of the slope in the x-direction is zero at that point, possibly indicating an inflection point in the x-direction if other conditions are met.